I agree that the sentence is unnecessarily wordy, but not ending a sentence with a preposition is an old, old mythrule, up with which we should no longer put! (Winston Churchill said something similar). This so-called rule has long outlived its usefulness. Read any of the great writers, and you will find their works abound with sentences ending in prepositions. Indeed, I would argue that putting prepositions in front of the relative pronouns "which" or "whom" is probably now far less favoured. Very 1920s.
Your sentence might sound a bit better if you removed the tautology of "currently ongoing".
Perhaps then reduce "going through" to "before".
"Placing someone in a home" smacks of dealing with elderly people, or with government foster care - whereas this is clearly about children staying with family members. I'd remove the word "home" altogether therefore and go with something like "which family member should have custody".
All these changes would give you:
"The case is currently before the courts to determine which family member should have custody of the children."
Quite by chance, this result has the added benefit of placating the preposition-sticklers amongst us. ;P